The TL/BL combo uses battery power so, no, the engine doesn't have to be running...or even in the bike. Only the HL circuit, i.e. HL bulb & high beam indicator are AC-powered and only while the engine is running. Just take things one logical step at a time and you'll get there. Wiring is intimidating, to most, the first time a problem has to be tracked-down. Keep in mind, the reason for this is the fact that there's not very much to be seen, no process that's visible in real time, unlike a mechanical issue like a punctured tires, oil leak, brake that doesn't work properly, etc. On the other hand, the system is simplicity itself, just multiplied by the number of individual leads. Now, take a deep breath and consider the methodology at it's most basic...one lead, or one device (switch, bulb, or horn) at a time is the only way to work on these electrical systems. That's the same for anyone, regardless of experience level...can't get any simpler than that. Where the anxiety level skyrockets is when that color-coded spaghetti is viewed in its "natural" and usually mangled, state.
Thus, the easiest way to inspect, and repair, a wire harness is while it's laying on your bench. IMO, there's a damned good chance that you'll see the problem. Armed with a schematic, a continuity tester, and patience, you will find any breaks or shorts. Equipped with a soldering gun, shrink tubing and electrical tape, it is possible to repair the wiring and do a first-rate job of it. I would, however, consider the price of a reproduction wire harness, before completing the reassembly of the bike. Less than $45 can bring a minty-new wire harness to your door...and that should be plug
& play.
